Re: Differentiating Implicitly-

Buffalo Books wrote:
Hello,

Could someone please help me to differentiate the following implicitly:

2x^2 y + sin y = 7

Thank you for your help in advance!

The good old-fashioned pre-Cauchy way:

d(2x^2 y + sin y) = 0;

d(2x^2 y + sin y) = d(2x^2 y) + d(sin y) =

2x^2 dy + 4xy.dx + cos(y).dy =

4xy.dx + (2x^2 + cos y)dy = 0.

Want derivative of y as a function of x?
Then dy/dx = -4xy / (2x^2 + cos y), and get rid of y in the RHS by means of the original relation 2x^2 y + sin y = 7. This requires solving a transcendental equation.

Want derivative of x as a function of y?
Then dx/dy = ... etc. This is easier than with y as a function of x.

All this is justified by the well-known definitions and theorems of differential calculus as founded by Cauchy. Think of the modern definition of differentiability, implicit-function theorems, differential 1-forms (gradients), and please keep in mind:
differential calculus is essentially about approximations of non-linear situations by their linear counterparts.

Johan E. Mebius
.